Express the following decimal as a rational number in simplest form:
The Correct Answer and Explanation is:
The repeating decimal 2.3‾2.\overline{3} can be expressed as the rational number 73\dfrac{7}{3} in simplest form.
Let’s walk through the method step by step to understand how this transformation works.
Start by letting a variable represent the repeating decimal: Let x=2.3333…x = 2.3333\ldots where the 3 continues indefinitely.
To eliminate the repeating part, multiply both sides of the equation by 10. This shifts the decimal point one place to the right: 10x=23.3333…10x = 23.3333\ldots
Now subtract the original x=2.3333…x = 2.3333\ldots from this equation:
10x−x=23.3333…−2.3333…10x – x = 23.3333\ldots – 2.3333\ldots
9x=219x = 21
Solving for xx, divide both sides by 9:
x=219x = \dfrac{21}{9}
This fraction can be simplified. Both 21 and 9 share a common factor of 3:
21÷39÷3=73\dfrac{21 \div 3}{9 \div 3} = \dfrac{7}{3}
So the decimal 2.3‾2.\overline{3} is equivalent to the rational number 73\dfrac{7}{3}.
Explanation
Repeating decimals are always rational, which means they can be expressed as the ratio of two integers. The strategy is to use a variable and algebraic manipulation to “trap” the repeating part. By multiplying by a power of 10, the repeating digits align, and subtracting cancels out the infinite sequence. What remains is a clean equation with only whole numbers, making it solvable using basic algebra.
Simplifying the resulting fraction ensures that we express it in the lowest terms possible. Understanding this technique deepens your appreciation for the structure of numbers and helps bridge the gap between decimal and fractional representations. It is especially useful in both theoretical math and real-world computations where precise values matter.
